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In the second half of the nineteenth century, a new interest in explosive chemical reactions, sudden release of energy in living beings, physical instabilities, and bifurcations in the solutions of differential equations drew the attention of some scholars. New concepts like triggering actions and guiding principles also emerged. Mathematicians, physicists, physiologists, and philosophers were attracted by this kind of phenomena since they raised a question about the actual existence of a strict determinism in science. In 1878 the mathematical physicist Joseph Boussinesq pointed out a structural analogy among physical instabilities, some essential features of living beings, and singular solutions of differential equations. These developments revived long-lasting philosophical debates on the problematic link between deterministic physical laws and free will. We find in Boussinesq an original and almost isolated attempt to merge mathematical, physical, biological, and philosophical issues into a complex intellectual framework. In the last decades, some philosophers of science rediscovered the connection between physical instabilities and determinism, both in the context of chaos theory, and in the debates on the Norton dome. I put forward a consistent historical reconstruction of the main issues and characters involved., (Copyright © 2014 Elsevier Ltd. All rights reserved.)

Discusses numerical methods to compute approximate solutions of a differential equation. Provides a Pascal program for the extended Euler-Heun method. (YP)

The study of the oscillatory behavior of a general class of neutral Emden-Fowler differential equations is the focus of this work. The main motivations for studying the oscillatory behavior of neutral equations are their many applications as well as the richness of these equations with exciting analytical issues. We obtained novel oscillation conditions in Kamenev-type criteria for the considered equation in the canonical case. We improve the monotonic and asymptotic characteristics of the non-oscillatory solutions to the considered equation and then utilize these characteristics to refine the oscillation conditions. We present, through examples and discussions, what demonstrates the novelty and efficiency of the results compared to previous relevant findings in the literature. In addition, we numerically represent the solutions of some special cases to support the theoretical results. [ABSTRACT FROM AUTHOR]

We study the Kneser-type oscillation criterion for a class of second-order half-linear functional dynamic equations on an arbitrary time scale utilizing the integral averaging approach and the Riccati transformation method. The results show an improvement in Kneser-type when compared to some known results. We provide some illustrative examples to demonstrate the significance of our main results.

In his lost notebook, Ramanujan presented unique categories of remarkable infinite series, known as the Ramanujan-type Eisenstein series. The objective of this paper is to generate various differential identities related to classical $ \eta $-functions and $ h $-functions with the help of the Ramanujan-type Eisenstein series. Moreover, we present particular identities that incorporate Eisenstein series of different levels and $ h $-functions, thereby establishing links between class one infinite series and $ h $-functions.

In this paper, numerical schemes based on non-polynomial splines, namely, spline in compression, tension, and adaptive spline, are constructed for singularly perturbed boundary value problems with mixed shifts. A convergence analysis is carried out on the proposed methods. A comparitive study of the results is performed on test problems and presented in the form of tables. Graphs are drawn to illustrate the behavior of the solution to the problems.

In this paper, we present a characterization of diagonal solutions for a class of linear matrix inequalities. We consider linear hybrid time-delay systems and explore the conditions under which these systems are positive and asymptotically stable. Specifically, we investigate the existence of positive diagonal solutions for a linear inequality when the system matrices are Metzler and nonnegative. Using various mathematical tools, including the Schur complement and separation theorems, we derive necessary and sufficient conditions for the stability of these systems. Our results extend existing stability criteria and provide new insights into the stability analysis of positive time-delay systems.

This study introduces a new approach to the development of generalized 1-parameter, 2-variable Hermite–Frobenius–Euler polynomials, which are characterized by their generating functions, series definitions and summation formulae. Additionally, the research utilizes a factorization method to establish recurrence relations, shift operators and various differential equations, including differential, integro-differential and partial differential equations. The framework elucidates the fundamental properties of these polynomials by utilizing generating functions, series definitions and summation formulae. The results of the study contribute to the understanding of the properties of these polynomials and their potential applications.

We present a novel framework for introducing generalized 3-variable 1-parameter Hermite-based Appell polynomials. These polynomials are characterized by generating function, series definition, and determinant definition, elucidating their fundamental properties. Moreover, utilizing a factorization method, we established recurrence relations, shift operators, and various differential equations, including differential, integrodifferential, and partial differential equations. Special attention is given to exploring the specific cases of 3-variable 1-parameter generalized Hermite-based Bernoulli, Euler, and Genocchi polynomials, offering insights into their unique features and applications.

We explored a class of quantum calculus boundary value problems that include fractional $ q $-difference integrals. Sufficient and necessary conditions for demonstrating the existence and uniqueness of positive solutions were stated using fixed point theorems in partially ordered spaces. Moreover, the existence of a positive solution for a boundary value problem with a Riemann-Liouville fractional derivative and an integral boundary condition was examined by utilizing a novel fixed point theorem that included a $ \mathfrak{a} $-$ \eta $-Geraghty contraction. Several examples were provided to demonstrate the efficacy of the outcomes.

Using comparison principles, we examine the asymptotic characteristics of a third-order nonlinear damped neutral differential equation. Our results substantially generalize numerous previously established results as well as drastically improving them. To illustrate the relevance and effectiveness of our results, we use numerical examples.

The present paper dealt with the study of solitons of Lorentzian Kähler space-time manifolds. In this paper, we have discussed different conditions for solitons to be steady, expanding, or shrinking in terms of isotropic pressure, the cosmological constant, energy density, nonlinear equations, and gravitational constant in pseudo-quasi-conformally flat and pseudo-$ W_8 $ flat Lorentzian Kähler space-time manifolds.

Picard's theorem is among the most striking results in complex analysis and plays a decisive role in the development of the theory of entire and meromorphic functions. In this paper, we give an equivalence between Picard's theorem and entire solutions of (i) difference equation Δ η f + a (z) P (f) = 0 in C and (ii) differential equation D k f + h (z) g (f (z)) = 0 in terms of total derivatives in C n . We prove two Picard-type theorems that generalize the results of Lu (Kodai Math J 26:221–229, 2003). In addition, examples have been exhibited to show that certain assumptions in the main result cannot be improved. [ABSTRACT FROM AUTHOR]

Suppose that M is a closed isotropic Riemannian manifold and that R1,...,Rm generate the isometry group of M. Let f1,...,fm ​ be smooth perturbations of these isometries. We show that the fi are simultaneously conjugate to isometries if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero. This extends a linearization result of Dolgopyat and Krikorian [Duke Math. J. {136}, 475-505 (2007)] from Sn to real, complex, and quaternionic projective spaces. In addition, we identify and remedy an oversight in that earlier work [ABSTRACT FROM AUTHOR]

In this study based on the accelerated over relaxation (AOR) method we make an iterative scheme for solving generalized Lyapunov matrix equation ... over complex or real matrices. Then we analyze the convergence of the new iterative method in detail. There have been discussions for the calculation of optimal parameters. Finally a numerical example is given to demonstrate the capability of the new method. [ABSTRACT FROM AUTHOR]

We deal with probabilistic numerical solutions for linear elliptic equations with Neumann boundary conditions in a Lipschitz domain, by using a probabilistic numerical scheme introduced by Milstein and Tretyakov based on new numerical layer methods. [ABSTRACT FROM AUTHOR]

The purpose of this study is to develop new iterative oscillation criteria for second‐order half‐linear advanced dynamic equations. These findings improve and extend recently established criteria for the same equation by several authors as well as encompass the existing classical criteria for associated ordinary dynamic equations. Some examples are given to show that the outcomes are accurate, practical, and adaptable. [ABSTRACT FROM AUTHOR]

We study locally maximal attractors of expanding dynamical systems. Our main result is a representation of these attractors with the help of topological Markov chains corresponding to the Markov partitions of these attractors, which allows us to describe the dynamics of system on them. Ya. G. Sinai was the first who constructed and used Markov partitions for Anosov's diffeomorphisms [Funk. Anal. Prilozh., 2, No 1, 64; No 3, 70 (1968); English translation:Funct. Anal. Appl., 2, No 1, 61; No 3, 245 (1968)]. Expanding endomorphisms regarded as the simplest representatives of endomorphisms were first studied by M. Shub [Amer. J. Math., 91, No 1, 175 (1969)]. To construct Markov partitions for expanding endomorphisms, we update Sinai's approach in the proper way. A more detailed historical overview can be found in the work by O. M. Sharkovsky [Ukr. Mat. Zh., 74, No. 12, 1709 (2023); English translation:Ukr. Math. J., 74, No. 12, 1950 (2023)]. In this work, Sharkovsky indicated that the methods used to prove the main results presented in [Dokl. Akad. Nauk SSSR, 170, No. 6, 1276 (1966); English translation:Sov. Math. Dokl., 7, No. 5, 1384 (1966)] were, in fact, published in the collection of papers "Dynamical systems and the problems of stability of solutions of differential equations" (1973) issued by the Institute of Mathematics of the Academy of Sciences of Ukraine. This collection is difficultly accessible and was never translated into English. Note that, in the indicated paper, these methods were applied to somewhat different objects. To the best of our knowledge, there is no information about publications of similar results. In view of the outlined history and importance of the described approach (based on Markov partitions and topological Markov chains) for the description of construction of the attractors, it seems reasonable to publish these results anew. [ABSTRACT FROM AUTHOR]

The basic reproduction, or reproductive number, is a useful index that indicates whether or not there will be an epidemic. However, it is also very important to determine whether an epidemic will eventually decrease and disappear or persist as an endemic. Different infectious diseases have different behaviors and mathematical models used to simulated them should capture the most important processes; however, the models also involve simplifications. Influenza epidemics are usually short-lived and can be modeled with ordinary differential equations without considering demographics. Delays such as the infection time can change the behavior of the solutions. The same is true if there is permanent or temporary immunity, or complete or partial immunity. Vaccination, isolation and the use of antivirals can also change the outcome. In this paper, we introduce several new models and use them to find the effects of all the above factors paying special attention to whether the model can represent an infectious process that eventually disappears. We determine the equilibrium solutions and establish the stability of the disease-free equilibrium using various methods. We also show that many models of influenza or other epidemics with a short duration do not have solutions with a disappearing epidemic. The main objective of the paper is to introduce different ways of modeling immunity in epidemic models. Several scenarios with different immunities are studied since a person may not be re-infected because he/she has total or partial immunity or because there were no close contacts. We show that some relatively small changes, such as in the vaccination rate, can significantly change the dynamics; for example, the existence and number of the disease-free equilibria. We also illustrate that while introducing delays makes the models more realistic, the dynamics have the same qualitative behavior.

This article delves into the behavior of solutions to a general class of functional differential equations that contain a neutral delay argument. This category encompasses the half-linear case and the multiple-delay case of neutral equations. The motivation to study this type of equation lies not only in the exciting analytical issues it presents but also in its numerous vital applications in physics and biology. We improved some of the inequalities that play a crucial role in developing the oscillation test. Then, we used an improved technique to derive several criteria that ensure the oscillation of the solutions of the studied equation. Additionally, we established a criterion that did not require imposing monotonic constraints on the delay functions and took into account their effect. We have supported the novelty and effectiveness of the results by analyzing and comparing them with previous results in the literature.

The focus of our investigation was on determining the existence of solutions for fractional differential equations (FDEs) of order $ 1 < \gamma\leq 2 $ involving the boundary conditions $ \kappa_{0}\phi(0)+\eta_{0}\phi(v) = \mu_{0} $, and $ \kappa_{1}\phi^{'}(0)+\eta_{1}\phi^{'}(v) = \mu_{1} $, for $ \kappa_i, \eta_i, \mu_i \in \mathbb{R}^{+} $. The existence results were based on the Schauder fixed point theorem and the nonlinear alternative of the Leray-Schauder type. Examples were provided to illustrate the results.

In this paper, we extend the block hybrid method with equally spaced intra-step points to solve linear and nonlinear third-order initial value problems. The proposed block hybrid method uses a simple iteration scheme to linearize the equations. Numerical experimentation demonstrates that equally spaced grid points for the block hybrid method enhance its speed of convergence and accuracy compared to other conventional block hybrid methods in the literature. This improvement is attributed to the linearization process, which avoids the use of derivatives. Further, the block hybrid method is consistent, stable, and gives rapid convergence to the solutions. We show that the simple iteration method, when combined with the block hybrid method, exhibits impressive convergence characteristics while preserving computational efficiency. In this study, we also implement the proposed method to solve the nonlinear Jerk equation, producing comparable results with other methods used in the literature.

This paper discusses the existence of solutions for fractional differential equations with nonlocal boundary conditions (NFDEs) under essential assumptions. The boundary conditions incorporate a point $ 0\leq c < d $ and fixed points at the end of the interval $ [0, d] $. For $ i = 0, 1 $, the boundary conditions are as follows: $ a_{i}, b_{i} > 0 $, $ a_{0} p(c) = -b_{0} p(d), \ a_{1} p^{'}(c) = -b_{1} p^{'}(d) $. Furthermore, the research aims to expand the usability and comprehension of these results to encompass not just NFDEs but also classical fractional differential equations (FDEs) by using the Krasnoselskii fixed-point theorem and the contraction principle to improve the completeness and usefulness of the results in a wider context of fractional differential equations. We offer examples to demonstrate the results we have achieved.

Copyright of HETS Online Journal is the property of Hispanic Educational Technology Services, Inc. (HETS) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The present investigation aims to examine the geometric properties of the normalized form of the combination of generalized Lommel–Wright function J λ , μ ν , m (z) : = Γ m (λ + 1) Γ (λ + μ + 1) 2 2 λ + μ z 1 − (ν / 2) − λ J λ , μ ν , m (z) , where the function J λ , μ ν , m satisfies the differential equation J λ , μ ν , m (z) : = (1 − 2 λ − ν) J λ , μ ν , m (z) + z (J λ , μ ν , m (z)) ′ with J ν , λ μ , m (z) = (z 2) 2 λ + ν ∑ k = 0 ∞ (− 1) k Γ m (k + λ + 1) Γ (k μ + ν + λ + 1) (z 2) 2 k for λ ∈ C ∖ Z − , Z − : = { − 1 , − 2 , − 3 , ... } , m ∈ N , ν ∈ C , and μ ∈ N 0 : = N ∪ { 0 } . In particular, we employ a new procedure using mathematical induction, as well as an estimate for the upper and lower bounds for the gamma function inspired by Li and Chen (J. Inequal. Pure Appl. Math. 8(1):28, 2007), to evaluate the starlikeness and convexity of order α, 0 ≤ α < 1 . Ultimately, we discuss the starlikeness and convexity of order zero for J λ , μ ν , m , and it turns out that they are useful to extend the range of validity for the parameter λ to λ ≥ 0 where the main concept of the proofs comes from some technical manipulations given by Mocanu (Libertas Math. 13:27–40, 1993). Our results improve, complement, and generalize some well-known (nonsharp) estimates. [ABSTRACT FROM AUTHOR]

We derive Melnikov type conditions for the persistence of periodic solutions in perturbed slowly varying discontinuous differential equations. In contrast to Battelli and Fečkan (Mathematics 9(19):2449, 2021) we assume that the unperturbed (frozen) equation has a family of periodic solutions depending on some parameters. The result of this paper are motivated by and extend a result in Wiggins and Holmes (SIAM J Math Anal 18:592–611, 1987) where the authors considered a two dimensional hamiltonian family of smooth systems depending on a scalar variable which is the solution of a singularly perturbed equation. [ABSTRACT FROM AUTHOR]

In this paper, the existence of positive solutions of the following third-order three-point boundary value problem with p-Laplacian {(Φp(u"(t)))' + f(t, u(t)) = 0, t (0, 1), u(0) = au(η), u(1) = au(η), u"(0) = 0, is studied, where Φp(s) = |s|p-2s, p > 1. By using the fixed point index method, we establish sufficient conditions for the existence of at least one or at least two positive solutions for the above boundary value problem. The main result is demonstrated by providing an example as an application. [ABSTRACT FROM AUTHOR]

Structuralism in philosophy of mathematics has largely focused on arithmetic, algebra, and basic analysis. Some have doubted whether distinctively structural working methods have any impact in other fields such as differential equations. We show narrowly construed structuralism as offered by Benacerraf has no practical role in differential equations. But Dedekind’s approach to the continuum already did not fit that narrow sense, and little of mathematics today does. We draw on one calculus textbook, one celebrated analysis textbook, and a monograph on the Navier–Stokes equation to show structural methods like Dedekind’s have long been central to differential equations, and have philosophically respectable ontology and epistemology. [ABSTRACT FROM AUTHOR]

Background. The primary element of the instrumentation for measuring the pressure of a gas-liquid medium is a sensor that supplies data on the pressure of the working medium, which determines the proper functioning of machines, mechanisms, and systems. Increasing the service life, reducing development time, and reducing the cost of sensors are one of the primary objectives. In this regard, mathematical modeling of the functioning of such systems plays an important role at the design stage of pressure measurement systems. To measure and control the pressure of the working gas-liquid medium in the combustion chambers of engines, a mechanical system “pipeline - pressure sensor” is used, in which, to reduce the effects of vibration accelerations and high temperatures, the sensor is connected to the engine via a pipeline and is located at some distance from it. The purpose of the work is to create a mathematical model of the “pipeline - pressure sensor” system and study it for the possibility of establishing a correspondence between the law of pressure change in the combustion chamber and the law of oscillation of the sensitive element of the pressure sensor. Materials and methods. To describe the movement of the working medium (in the ideal gas model), a nonlinear model of fluid and gas mechanics is used, under the assumption that the working medium is compressible. To describe the dynamics of the sensitive element of the sensor, a model is used, the basis of which is an ordinary differential equation that describes the oscillatory process of a single-mass system. Under these assumptions, a mathematical model of the mechanical system “pipeline – pressure sensor” was constructed. To solve the corresponding problem, the formulation of which contains a nonlinear partial differential equation, numerical and analytical solution methods based on the Galerkin method are proposed. Results. A nonlinear mathematical model of a system for measuring pressure in gas-liquid media has been developed. For the corresponding initial-boundary value problem, based on the Galerkin method, a method is proposed that makes it possible to reduce its study to solving the Cauchy problem for a system of ordinary differential equations. A numerical experiment is carried out and examples of calculating the dynamics of the sensor's sensitive element are presented. Conclusions. The proposed mathematical model makes it possible to determine the law of change in the deviation of the sensor's sensitive element depending on the law of change in pressure in the combustion chamber. The research results are intended for use at the design stage of pressure measurement systems.

A projective space in which a linear group acts ineffectively allows one to construct the corresponding space with a Cartan projective connection. We show that the structural equations of the Cartan space allow one to obtain differential equations for the components of the projective curvaturetorsion tensor. This tensor contains the torsion tensor, the extended torsion tensor, and the affine curvature-torsion tensor. An analog of the Bianchi identities is found. An algorithm for constructing structural equations of the space with a Cartan projective connection is formulated. Using a generalized algorithm, we construct the structural equations of the planar space with a projective connection, whose special cases are the ruled space with an Akivis projective connection, the point space with a Cartan projective connection, and its dual hyperplanar space with a projective connection. We also prove that the curvature-torsion tensor of the plane space with projective connection has three subtensors, one of which is an analog of the torsion tensor of the Cartan space. Keywords and phrases: space with a Cartan projective connection, curvature-torsion tensor, analog of Bianchi identities, ruled space with a projective connection, planar space with a projective connection. AMS Subject Classification: 53B10, 1. Space with a Cartan projective connection. Consider the n-dimensional projective space [P.sub.n] with a homogeneous moving frame {[A.sub.I']} with the derivation formulas [Please download the PDF to view the [...]

We study embedding operators on Orlicz-Sobolev spaces generated by the composition rule. Using the composition operators we consider embeddings of the Orlicz-Sobolev spaces into weighted Orlicz spaces. Bibliography: 46 titles., 1 Introduction The Sobolev embeddings theory plays a significant role in the study of existence and regularity of weak solutions of partial differential equations [1, 2]. The classical method of [...]

We consider an integro-differential fractional order equation describing one-dimensional eigenoscillations of a homogeneous viscoelastic medium. We show that the spectrum of this equation consists of N sequences of eigenvalues, where N = 4 or N = 5. Elements of these sequences are squared roots of equations of degree N. We study their limit behavior. We prove that there are only two sequences consisting of points with negative real part. Bibliography: 9 titles., The paper is devoted to the spectral analysis of an integro-differential equation containing the Gerasimov-Caputo fractional time-derivative of order [alpha] and the fractional exponential Rabotnov kernel [R[alpha]-1](-[beta],t), where [alpha] = [...]

The existence and uniqueness theorems of the solution to the boundary-value problem for a parabolic-hyperbolic fractional-order equation with the gluing condition are proved. Keywords and phrases: fractional derivative, loaded equation, differential equation, integral equation. AMS Subject Classification: 34K37, 1. Introduction. The first fundamental research in the theory of loaded equations is due to A. M. Nakhushev. In [14, 16], the general definition of a loaded equation and a [...]

This article presents the mathematical formulations of transient heat conduction problems corresponding to the models of classical heat conduction using the Fourier law and generalized heat conduction based on the Cattaneo–Vernotta–Lykov law (Maxwell–Cattaneo model), as well as the generalized Green–Nagdy type II and III models. The Fourier transforms in spatial coordinates and the Laplace transforms in time were used to obtain the fundamental solutions of the equations of the Maxwell–Cattaneo and Green–Nagdy type II and III models of classical and generalized heat conduction. The results were displayed graphically and analyzed. Differences between the considered heat conduction models were shown, and suggestions for their practical application were given.

In the present study, we introduce a collocation approach utilizing quintic B-spline functions as bases for solving systems of Lane Emden equations which have various applications in theoretical physics and astrophysics. The method derives a solution for the provided system by converting it into a set of algebraic equations with unknown coefficients, which can be easily solved to determine these coefficients. Examining the convergence theory of the proposed method reveals that it yields a fourth-order convergent approximation. It is confirmed that the outcomes are consistent with the theoretical investigation. Tables and graphs illustrate the proficiency and consistency of the proposed method. Findings validate that the newly employed method is more accurate and effective than other approaches found in the literature. All calculations have been performed using Mathematica software.

In the setting of 2-uniformly convex Banach spaces, we prove the existence of solutions for a variant of the implicit state-dependent convex sweeping processes. Our approach is based on a differential equation associated with the generalized projection operator.

In this article, we have tried to introduce one of the most important topics in the subject of dynamical systems, namely the Melnikov function, in simple language. Melnikov function is one of the tools that can express the effect of the small perturbation on the homoclinic orbits. This issue is also related to breaking a homoclinic orbit under the effect of some small perturbations. When a small perturbation occurs in a dynamical system, some dynamical behaviors of the system may change. Here, we try to explain how to compute the formula of this function fluently. Therefore, in the first part, we will introduce some preliminary concepts and properties, and then in the second part, we will describe the construction of the Melnikov function. Finally, by stating the results of the fundamental matrix solutions and then using them, we will construct the Melnikov function near a homoclinic or hetroclinic orbit. 1- IntroductionOne of the fundamental issues that have recently attracted the attention of researchers in the field of theoretical research is the issue of dynamical systems. The phenomenon of touching stable and unstable manifolds is an example of behavior in a dynamical system that results in the existence of special solutions. If the stable and unstable curves of an equilibrium point touch each other, then the solution is homoclinic, and if the stable curves of one equilibrium point intersect with the unstable curve of another equilibrium point, then we will have a heteroclinic solution. Investigating the existence of such solutions for a specific dynamical system has always been one of the important issues, and therefore many people have done good research in this field, for example, some examples of such research can be found in [8, 4,5] observed. In addition, many researchers have tried to solve certain cases of Hilbert's 16th problem using the Melnikov function. As an example, we can refer to [2, 5, 6]. The subject we will discuss in this article is related to the breaking of a homoclinic orbit. When a perturbation occurs in a dynamic system, some dynamical behaviors of the system may change; One of these changes can be the breaking of the homoclinic orbit in the phase space. One of the tools that can express the effect of small perturbations on the homoclinic orbit for us is the Melnikov function, which in this article we try to explain how to make fluently. To continue our discussion more easily, it is necessary to familiarize the reader with a series of basic concepts, so in the second part we will introduce some preliminary concepts, and then in the third part we will describe the construction of the Melnikov function. 3- Main ResultsWe consider the differential equation $ \dot{x} = f(x,\mu) $ where $ x \in \mathbb{R}^2 $, $ \mu \in \mathbb{R} $ and $f$ is at least $C^2$. Suppose there are two hyperbolic saddles $ p_{-}(\mu) $ and $ p_{+}(\mu) $. Suppose that for $ \mu = 0 $ there is a solution of this equation in the form of $ x_{\ast}(t) $ such that\begin{align*}\lim_{t\rightarrow - \infty} x_{\ast}(t) = p_{-}(0),~~~~~~ \lim_{t\rightarrow + \infty} x_{\ast}(t) = p_{+}(0) \end{align*} Our goal is to check that when $ \mu $ changes, how this connection (that is, the saddle connection between $p_{-}(t)$ and $p_{+}(t)$) is broken.\\We put $ x_{\ast}(0) = x_{0} $. The velocity vector $ x_{\ast}(t) $ at $ t = 0 $ is:\begin{align*}\dot{x}_{\ast}(0) = f(x_{0},0) = (f_{1}(x_{0},0),f_{2}(x_{0},0 )).\end{align*}If we put\begin{align*}u_{0} = \frac{1}{\Vert f(x_{0},0)\Vert^{2}}(-f_{2}(x_{0},0) , f_{1 }(x_{0},0)), \end{align*}then obviously, the vector $ u_{0} $ is perpendicular to $ f(x_{0},0) $. We consider a line segment $ \Sigma $ passing through $ x_{0} $ and in the direction of $ u_{0} $. Therefore, $ \Sigma $ with the formula $ x = x_{0} + \xi u_{0} $ where $ \vert \xi \vert < \alpha $ for some $ \alpha>0$. Suppose $ x_{-}(t,\mu) $ is a solution of $ \dot{x} = f(x,\mu) $ such that$$1)~~x_{-}(0,\mu) \in \Sigma,$$$$2)~~\lim_{t\rightarrow -\infty}x_{-}(t,\mu) = p_{-}(\mu),$$$$3)~~x_{-}(t,0) = x_{\ast}(t).$$This answer is on the unstable manifold $ p_{-}(\mu) $.Similarly, suppose that $ x_{+}(t,\mu) $ is a solution of $ \dot{x} = f(x,\mu) $ such that$$1)~~ x_{+}(0,\mu) \in \Sigma,$$$$2)~~\lim_{t\rightarrow +\infty}x_{+}(t,\mu) = p_{+}(\mu),$$$$3)~~x_{+}(t,0) = x_{\ast}(t).$$This solution is also placed inside the stable manifold $ p_{+}(\mu) $ (pay attention to the Figures 1 and 2). We define the separation function as follows:\begin{align*}s(\mu) = \xi_{-}(\mu) - \xi_{+}(\mu). \end{align*}We put $ \psi_{0} = (-f_{2}(x_{0},0),f_{1}(x_{0},0)) $ then $\dot{x}_{*}(t)=(\dot{x}_{*1}(t),\dot{x}_{*2}(t))$ is a solution of the equation $\dot{v}(t)=A(t)v$ and so,$$ \psi(t) = exp\left(-\int_{0}^{t} div f(x_{\ast}(s),0)ds\right)\left(-\dot{x}_{\ast2}(t), \dot{x}_{\ast1}(t)\right)$$is a solution of the adjoint equation $\dot{w}=-wA(t)$, with the initial condition$$\psi(0)=(-\dot{x}_{*2}(0),\dot{x}_{*1}(0))=(-f_2(x_0,0),f_1(x_0,0))=\psi_0,$$Since we have $\psi_0u_0=1,$ it follows that:\begin{align*}\frac{d \xi_{\pm}}{d\mu}(0) = \psi_{0}\frac{d \xi_{\pm}}{d\mu}(0)u_{0} = \psi_{0}\frac{\partial x_{\pm}}{\partial \mu}(0,0).\end{align*}Also $ \frac{\partial x_{\pm}}{\partial \mu}(t,0) $ is an answer of the following equation:\begin{align}\label{b50}\dot{v} = D_{x}f(x_{\ast}(t),0)v + \frac{\partial f}{\partial \mu}(x_{\ast}(t) ,0).\end{align}Let's assume that the state transition matrix of the homogeneous linear system $ \dot{v} = D_{x}f(x_{\ast}(t),0) v $ is $ \phi(t,s) $. Now, according to the parameter change formula, we have:\begin{align*}\frac{\partial x_{+}}{\partial\mu}(0,0)=\phi(0,T)\frac{\partial x_{+}}{\partial\mu}(T, 0)-\int_{0}^{T}\phi(0,s)\frac{\partial f}{\partial\mu}(x_{\ast}(s) ds, 0)\end{align*}And\begin{align*}\frac{\partial x_{-}}{\partial\mu}(0,0)=\phi(0,-T)\frac{\partial x_{-}}{ \partial\mu}(-T, 0)-\int_{-T}^{0}\phi(0,s)\frac{\partial f}{\partial\mu}(x_{\ast} (s), 0) ds\end{align*}So\begin{eqnarray}\label{b51}\frac{d \xi_{-}}{d \mu}(0) = \psi_0 \frac{\partial x_{-}}{\partial\mu}(0,0) &=& \psi (0)(\phi(0,-T)\frac{\partial x_{-}}{\partial \mu}(-T,0) + \int_{-T}^{0}\phi( 0,s)\frac{\partial f}{\partial\mu}(x_{\ast}(s),0)ds)\\&=& \psi(-T)\frac{\partial x_{-}}{\partial \mu}(-T,0) + \int_{-T}^{0}\psi(s) \frac{\partial f}{\partial \mu}(x_{\ast}(s),0)ds. \end{eqnarray}Now, if $t\rightarrow\infty$ then:\begin{align*}\lim_{T\rightarrow +\infty}\psi(-T)\frac{\partial x_{-}}{\partial\mu}(-T,0) = 0. \end{align*}So\begin{align*}\frac{d \xi_{-}}{d \mu}(0) = \int_{-\infty}^{0} \psi(s)\frac{\partial f}{\partial \mu}( x_{\ast}(s),0)ds. \end{align*}Similarly\begin{align*}\frac{d \xi_{+}}{d \mu}(0) = \int_{0}^{+\infty} \psi(s)\frac{\partial f}{\partial \mu }(x_{\ast}(s),0)ds. \end{align*}So\begin{align*}s'(0) = \xi'_{-}(0) - \xi'_{+}(0) = \int_{-\infty}^{\infty} \psi(s)\frac {\partial f}{\partial \mu}(x_{\ast}(s),0).ds \end{align*} The recent integral is called the Melnikov integral if the function $f$ is dependent on $t$, that is, the right side of the differential equation is $f(x,\mu, t)$, then the value of Melnikov integral will also depend on $t$. 3- ConclusionsThis article presents a unique perspective on constructing the Melnikov integral, different from what is typically found in textbooks. The problem's assumptions are presented in a way that accounts for both homoclinic and heteroclinic states. By following the method outlined in the article, interested readers can calculate Melnikov's integral formula in the time-dependent mode by considering the time-dependent perturbation on the system.

Let $m \ge 2$ and $a, b, c > 0$. We consider the existence and uniqueness of solutions for the fourth order iterative boundary value problem, \begin{equation*} x^{(4)}(t) = -f(t, x(t), x^{[2]}(t), \dots, x^{[m]}(t)), \qquad -a \le t \le a \end{equation*} \noindent where $x^{[2]}(t)$ =$x \big(x(t) \big)$ and for $j = 3,\dots, m$, $x^{[j]}(t)$ = $x \big( x^{[j-1]}(t) \big)$, with solutions satisfying one of the following sets of conjugate boundary conditions: \begin{equation*} \begin{aligned} x(-a) &= -a, &\quad x'(-a)& = b,& \quad x''(-a) &= c,& \quad x(a) &= a,\\ x(-a) &= -a, &\quad x(a) &= a,& \quad x'(a) &= b,& \quad x''(a)& = c. \end{aligned} \end{equation*} The main tool used is the Schauder fixed point theorem.

Differential equations are a fundamental and versatile mathematical tool that finds widespread application across diverse academic disciplines, from physics and biology to economics and engineering. The primary objectives of this report are to demonstrate the application of differential equations in stellar evolution, construct a mathematical model to demonstrate nuclear reactions in a star, and illustrate energy transport within a star. Triangulation was used to prepare this report, with literature studies being the primary method. This study includes several documents and field data analyzed using qualitative research. Through research and observations, two hypothetical case studies illustrate the indispensable application of differential equations in modeling energy transport and nuclear reactions within stars through which the value of luminosity was calculated in a particular star due to both radiative energy transport and convective energy transport while in another star, the helium abundance in the core was estimated to approach a value of 1.195*1077. These differential equations are not only limited to the growth of a lead but also have broader applications that are essential for understanding the chemical composition of the universe and its prolonged evolution. The report also underscores the enduring importance of differential equations in advancing our understanding of the cosmos and their vital role in space exploration and technological innovations.

Economic growth modeling is one of the methods a government can use to formulate appropriate economic policies to improve the prosperity of its people. Differential equations and stochastic models play a major role in studying economic growth. This article aims to conduct a literature review on the use of differential equations in relation to stochastics to model economic growth. In addition, this article also discusses the use of differential and stochastic equations in economic growth models in Indonesia. This study involves searching for and selecting articles to obtain a collection of research works relevant to the application of differential and stochastic equations to economic growth models, supported by bibliometric analysis. The results of this literature review show that there is still little research discussing economic growth models using differential equations combined with stochastic models, especially those applied in Indonesia. While the application of these models remains relatively limited, their potential to offer deeper insights into the complex dynamics of economic growth is undeniable. By further developing and refining these models, we can gain a more comprehensive understanding of the factors driving growth and the potential implications of various economic policies. This will ultimately equip policy-makers with a more powerful analytical tool for making informed decisions.

This article highlights the oscillatory properties of second-order Emden–Fowler delay differential equations featuring sublinear neutral terms and multiple delays, encompassing both canonical and noncanonical cases. Through the proofs of several theorems, we investigate criteria for the oscillation of all solutions to the equations under study. By employing the Riccati technique in various ways, we derive results that expand the scope of previous research and enhance the cognitive understanding of this mathematical domain. Additionally, we provide three illustrative examples to demonstrate the validity and applicability of our findings.

Corruption is a global problem that affects the fair distribution of wealth of every country to different degrees and represents a problem to be solved to prevent the diversion and waste of resources. Among the different efforts to first measure it and later reduce it by proposing strategies, there exist a variety of indices, such as the corruption perception index, and other related issues, such as the global impunity index, the laxness of anticorruption policies, etc., which are computed for different countries worldwide. Based on these indices, we propose a model for corruption using a system of ordinary differential equations, considering anticorruption policies. Those three factors were identified after analyzing the phenomenon and available data, particularly for Mexico. Also, we fit it to the reported data of this country and perform simulations expecting to predict the short term, and performed a sensitivity analysis. The model is capable of reproducing the observed oscillatory behavior of the phenomenon. The model fit can still be improved by including the data for the anticorruption policies, which were only studied for different scenarios. Moreover, the model is susceptible to application in other countries, as long as data are available, and then provides a computational tool to predict and visualize the effect of appropriate public policies to fight corruption.

In this paper, we introduce the concept of generalized Fourier series, generated by the p-trigonometric functions, namely cosp and sinp, recently introduced related to the generalized complex numbers systems. The aim of this study is to represent a periodic signal as a sum of p-sine and p-cosine functions. In order to achieve this, we first present the integrals of the product of the same or different family of p-trigonometric functions over the full period of these functions to understand the orthogonality properties. Next, we use these integrals to derive the coefficients of the generalized p-Fourier series along with a few examples. The generalized Fourier series can be used to expand an arbitrary forcing function in the solution of a non-homogeneous linear ordinary differential equation (ODE) with constant coefficients.

This paper investigates the oscillatory behavior of solutions to fourth-order functional differential equations (FDEs) with multiple delays and a middle term. By employing a different comparison method approach with lower-order equations, the study introduces enhanced oscillation criteria. A key strength of the proposed method is its ability to reduce the complexity of the fourth-order equation by converting it into first- and second-order forms, allowing for the application of well-established oscillation theories. This approach not only extends existing criteria to higher-order FDEs but also offers more efficient and broadly applicable results. Detailed comparisons with previous research confirm the method’s effectiveness and broader relevance while demonstrating the feasibility and significance of our results as an expansion and improvement of previous results.

Separation of variables is a common method for producing an analytical based solution to partial differential equations. Despite the wide application of this method, often the physical phenomena described by the differential equations are not adequately involved in the discourse over the appropriate methods to solve a given problem, particularly in mathematics curricula. However, as mathematics is the tool to better understanding of the physical world, the meaning of the differential equation, boundary conditions, and initial conditions cannot be detached from the methods used to solve the differential equations. Failure to recognize the physical conditions being studied can lead to solution methods that are invalid or unphysical. This paper demonstrates how awareness of the physical nature of the system being investigated and its relationship to the mathematics can guide the selection of the relevant solution methods. To illustrate the importance of the comprehension of the physical meaning behind the mathematical equations and representations and the need to avoid rote application a solution technique, the logic behind the selection of the appropriate solution techniques for the one-dimensional transient heat conduction equation is considered under different imposed conditions which lead to different trends in system operation. [ABSTRACT FROM AUTHOR]

In this article, we investigate some properties of meromorphic solution of the following differentialdifference equation ∑j=0n cjf (z + j) + ∑j=0n ljf(j) (z + j) + ∑j=0m djf(j) (z) -- c = (f (z + η) -- f (z) -- b)eQ, where n,m ≥ 1 are two integers, ηis a constant with |η| ≠ n, Q is a polynomial. We study the growth of solutions of a more general differential-difference equation given by Lü et al [Rest Math 74 (2019):1-18]. Meantime we obtain the relation between deg Q and p(f). [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and stability of solutions in connection to a non-local multiterm boundary value problem (BVP) with differential equations equipped with the Riemann-Liouville (RL) fractional derivative in the sense of Atangana-Baleanu of variable-order. The results about the existence property are investigated and proved via Krasnoselskii's fixed point theorem. Note that all theorems in the present research are studied based on piece-wise constant functions defined on generalized intervals. We shall convert our main BVP with the RL-fractional derivative of the Atangana-Baleanu type of variable-order to an equivalent BVP of constant order of the RL-Atangana-Baleanu derivative. In the next step, we examine the Ulam-Hyers stability for the supposed variable-order RL-Atangana-Baleanu BVP. Finally, we provide some examples to validate that our results are applicable. [ABSTRACT FROM AUTHOR]